math 374 graphs. topics cartesian plane methods of graphing intercept slope scale first quadrant...
TRANSCRIPT
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Math 374
Graphs
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Topics
Cartesian Plane Methods of Graphing Intercept Slope Scale First Quadrant Inequality Graphs Region
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Cartesian Plane
Named after Rene Deo Cartes a french mathematician
Also a philosopher “I think therefore I am” His goal was to create a “picture” that could
show a relationship between two variables. We have one for one variable – the number line.
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Notes
We recall
- 2 -1 0 1 2 3
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Some Facts
We only need two points to draw a straight line
The point where a graph crosses or touches the x axis is called the x intercept
It is found by substituting y = 0 The point where a graph crosses or touches
the y axis is called the y intercept It is found by substituting x = 0
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Intercept Method
Calculate both intercepts. Place on graph and join
Example #1: y = 2x – 6 X intercept (y = 0) 0 = 2x – 6 -2x = - 6 x = 3
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Intercept Method
Now find Y intercept Example #1: y = 2x – 6 Y intercept (x = 0) y = 2 (0) – 6 y = -6
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Finding X and Y Intercept
Example #2: 5x – 3y = 15 x int (y = 0) 5x – 3(0) = 15 5x = 15 x = 3 y Int (x = 0) 5(0) – 3y = 15 -3y = 15 y = -5
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Drawing on Graph
Now that you know the x & y intercept, you have two points and now can draw the straight line… do it!
Practice plotting with other points…
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1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
yPlottingPlotting
. C (-4, -4)
.
.
.. B (-4, 2)
D (2, -2)
(A 3, 1)
Q1 (+,+)
Q4 (+,-)Q3 (-,-)
Q2 (-,+)A (3,1)
B (-4,2)
C (-4, -4)
D (2, -2)
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Standard Form Method
All straight lines have a y intercept and a slant called a slope.
If the relationship is in standard form we can write it…
y = m x + b
Slope Y intercept
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Y Int
Slant
Slant
Y Int
Identifying Slant and Slope
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Standard Form
Recall y = mx + b
Dependent Variable (DV)
SlopeIndependent Variable (IV)
Y Intercept or Starting Value
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Relationship of y & b
It is easy to see how b is the y intercept; we substitute x = 0
x = 0 y = m(0) + b y = b
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Rise, Run & Slope
Slope
Rise
Run
Rise
Run
Slope
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Understanding the Slope
If m or the slope is 2 this means a rise of
2 and a run of 1 (2 can be written as 2 )
1 If m = - 5, this means a rise of -5 and right 1 If m= -2 this means rise of -2 right 3
3
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Understanding the Slope
Consider m = -3 4 What is the rise and what is the run? Suggest to put the negative sign on the top
to clarify (rise of -3) Numerator always rise (could go up or down) Denominator always run (right only)
RiseRun
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Consider y = 2x + 3
What is the slope, rise, run and y intercept? We have a slope 2 2 can be written as 2
1
Rise of 2
Run of 1
y intercept of 3 (y = b)
Plot on graph paper the following…
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1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Ex#1: y=2x+3Ex#1: y=2x+3
0,3
(1,5)
Question: Draw this line
What is the y intercept?
What is the slope
What does the slope mean?
Where can you plot the y intercept?
Up 2, Right 1
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Example #2 y = -5 x + 1 7
What is the y intercept, slope? Rise and run? Y intercept is 1 Slope is -5/7 Rise is – 5 Run is 7 Plot on graph (put it on graph paper)
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Example #3 y = x
What is the y intercept, slope, rise and run? y intercept = 0 (y int let x = 0) Slope = 1 Rise of 1 Run of 1 Plot on graph
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Example #4: 3x – 4y = 12
What is the y intercept, slope, rise and run? Must put in standard form -4y = - 3x + 12 y = 3x – 3 4 y intercept = -3 Slope ¾ Rise of 3, run of 4 Plot on graph
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Graphing with Scale
Scale is mostly used to make sure your graph can be seen
Consider y = 2x + 100
3
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Ex#5 y=2x+100 3Ex#5 y=2x+100 3
(0,100)
(300,300)
500
x
y
You can put 500 along the x axis which means each hash mark is 100
Y intercept?
Slope?
How will you measure m = 2/3?
Note slope is a ratio so scale does not effect it
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Ex. #5 200x + 300y = 120000
300y = - 200x + 120000 Y = -2x + 400
3 Plot it Do #4 on stencil use form C
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1st Quadrant
There will be times when you will need to put the graph only in the 1st quadrant
The problem only exists when the y intercept is negative
In that case, work with the x intercept (sub y = 0)
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Consider y = 2x – 5 3
Show how the graph intersects in the 1st quadrant Notice that b is negative. In those cases, work with x int (let y = 0) 0 = 2x – 5
3 0 = 2x – 15 -2x = -15 x = 7.5
Stencil: Do #5
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Inequality Graphs
The straight line of the graph divides the plane into two regions
One side will be greater than, one side less than
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The Trick in Standard Form
If greater then shade above > If less then shade below < If equal then solid line If not equal then dotted line
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Ex y > x + 3 Ex y > x + 3
5
x
y
Y intercept?
Slope?
Step 1: Draw Line
m = 1 (up 1, right 1)
Dotted Line or solid?
Shade above or below?
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Ex y < x + 3 Ex y < x + 3
5
x
y
Y intercept?
Slope?
Step 1: Draw Line
m = 1 (up 1, right 1)
Dotted Line or solid?
Shade above or below?
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Ex 5x - 10y < 30 Ex 5x - 10y < 30
5
x
y
y intercept?
Slope?
Step 1: Put in Standard Form
m = 1 (up 1, right 2)Dotted Line or solidline?
Shade above or below?
-10y < - 5x + 30
y > 1x – 3
2
Do #6 in C
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Point of Intersection
• If we have two graphs, we create four regions
1
3
4 2
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Consider y > 3x – 5 y < -2x + 5 Consider y > 3x – 5 y < -2x + 5
5
x
y
Draw lines… one at a time
Slope?
Hint… with 2 lines, use arrows at first instead of shading
m = 3 (3 up, right 1)
Dotted Line or solid?
Shade above or below?
y intercept of 1st? 2nd line…
y int? Slope?
Dotted / solid?
Use arrows
Above or Below?
Shade where they intersect!
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Find POI (Point of Intersection), you can also use equations
y = 3x – 5 y = -2x + 5 3x – 5 = -2x + 5 5x = 10 x = 2 x = 2 y = 3(x) – 5 y = 3 (2) – 5 y = 1 POI (2, 1)
Do 7 in E
Finish Study Guide